Chisquare {stats}R Documentation

The (non-central) Chi-Squared Distribution


Density, distribution function, quantile function and random generation for the chi-squared (chi^2) distribution with df degrees of freedom and optional non-centrality parameter ncp.


dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)


x, q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number required.
df degrees of freedom (non-negative, but can be non-integer).
ncp non-centrality parameter (non-negative). Note that ncp values larger than about 1417 are not allowed currently for pchisq and qchisq.
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


The chi-squared distribution with df= n > 0 degrees of freedom has density

f_n(x) = 1 / (2^(n/2) Gamma(n/2)) x^(n/2-1) e^(-x/2)

for x > 0. The mean and variance are n and 2n.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = λ has density

f(x) = exp(-lambda/2) SUM_{r=0}^infty ((lambda/2)^r / r!) dchisq(x, df + 2r)

for x >= 0. For integer n, this is the distribution of the sum of squares of n normals each with variance one, λ being the sum of squares of the normal means; further,
E(X) = n + λ, Var(X) = 2(n + 2*λ), and E((X - E(X))^3) = 8(n + 3*λ).

Note that the degrees of freedom df= n, can be non-integer, and for non-centrality λ > 0, even n = 0; see Johnson, Kotz and Balakrishnan (1995, chapter 29).


dchisq gives the density, pchisq gives the distribution function, qchisq gives the quantile function, and rchisq generates random deviates.


Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, Kotz and Balakrishnan (1995). Continuous Univariate Distributions, Vol 2; Wiley NY;

See Also

A central chi-squared distribution with n degrees of freedom is the same as a Gamma distribution with shape a = n/2 and scale s = 2. Hence, see dgamma for the Gamma distribution.


dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above

x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df=2), dexp(x, 1/2))
all.equal(pchisq(x, df=2), pexp(x, 1/2))

## non-central RNG -- df=0 is ok for ncp > 0:  Z0 has point mass at 0!
Z0 <- rchisq(100, df = 0, ncp = 2.)

## Not run: 
## visual testing
## do P-P plots for 1000 points at various degrees of freedom
L <- 1.2; n <- 1000; pp <- ppoints(n)
op <- par(mfrow = c(3,3), mar= c(3,3,1,1)+.1, mgp= c(1.5,.6,0),
          oma = c(0,0,3,0))
for(df in 2^(4*rnorm(9))) {
  plot(pp, sort(pchisq(rr <- rchisq(n,df=df, ncp=L), df=df, ncp=L)),
       ylab="pchisq(rchisq(.),.)", pch=".")
  mtext(paste("df = ",formatC(df, digits = 4)), line= -2, adj=0.05)
mtext(expression("P-P plots : Noncentral  "*
                 chi^2 *"(n=1000, df=X, ncp= 1.2)"),
      cex = 1.5, font = 2, outer=TRUE)
## End(Not run)

[Package stats version 2.2.1 Index]